I have BW installed on Chrome works fine, however I like a logo pinned to task bar. I pinned to taskbar, but every time I need to log in with Mpw, even after I change settings. I like it to be accessible when laptop is not locked.
You can go to Settings > Account Security, and configure the “Vault Timeout” options as follows:
This is not recommended (as it significantly weakens your security), but it would achieve the results you are looking for.
P.S. I corrected a typo in your post title (“lock” → “log”), and I changed your app:
tag from app:desktop
to app:browser
(since your post is about the browser extension, not the desktop app).